| L(s) = 1 | + (1.56 + 0.615i)2-s + (−1.65 − 0.519i)3-s + (0.614 + 0.570i)4-s + (2.65 + 1.80i)5-s + (−2.27 − 1.83i)6-s + (2.29 + 1.31i)7-s + (−0.849 − 1.76i)8-s + (2.46 + 1.71i)9-s + (3.04 + 4.46i)10-s + (−0.541 − 3.58i)11-s + (−0.719 − 1.26i)12-s + (−4.31 + 3.43i)13-s + (2.78 + 3.47i)14-s + (−3.44 − 4.36i)15-s + (−0.371 − 4.95i)16-s + (−4.48 − 1.38i)17-s + ⋯ |
| L(s) = 1 | + (1.10 + 0.435i)2-s + (−0.954 − 0.299i)3-s + (0.307 + 0.285i)4-s + (1.18 + 0.808i)5-s + (−0.927 − 0.747i)6-s + (0.867 + 0.497i)7-s + (−0.300 − 0.623i)8-s + (0.820 + 0.571i)9-s + (0.963 + 1.41i)10-s + (−0.163 − 1.08i)11-s + (−0.207 − 0.363i)12-s + (−1.19 + 0.953i)13-s + (0.745 + 0.929i)14-s + (−0.888 − 1.12i)15-s + (−0.0929 − 1.23i)16-s + (−1.08 − 0.335i)17-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 147 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & (0.876 - 0.481i)\, \overline{\Lambda}(1-s) \end{aligned}\]
Particular Values
| \(L(1)\) |
\(\approx\) |
\(1.57246 + 0.403922i\) |
| \(L(\frac12)\) |
\(\approx\) |
\(1.57246 + 0.403922i\) |
| \(L(\frac{3}{2})\) |
|
not available |
| \(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
|---|
| bad | 3 | \( 1 + (1.65 + 0.519i)T \) |
| 7 | \( 1 + (-2.29 - 1.31i)T \) |
| good | 2 | \( 1 + (-1.56 - 0.615i)T + (1.46 + 1.36i)T^{2} \) |
| 5 | \( 1 + (-2.65 - 1.80i)T + (1.82 + 4.65i)T^{2} \) |
| 11 | \( 1 + (0.541 + 3.58i)T + (-10.5 + 3.24i)T^{2} \) |
| 13 | \( 1 + (4.31 - 3.43i)T + (2.89 - 12.6i)T^{2} \) |
| 17 | \( 1 + (4.48 + 1.38i)T + (14.0 + 9.57i)T^{2} \) |
| 19 | \( 1 + (-0.711 + 0.410i)T + (9.5 - 16.4i)T^{2} \) |
| 23 | \( 1 + (0.734 + 2.38i)T + (-19.0 + 12.9i)T^{2} \) |
| 29 | \( 1 + (0.788 - 0.180i)T + (26.1 - 12.5i)T^{2} \) |
| 31 | \( 1 + (0.958 + 0.553i)T + (15.5 + 26.8i)T^{2} \) |
| 37 | \( 1 + (6.79 - 6.30i)T + (2.76 - 36.8i)T^{2} \) |
| 41 | \( 1 + (-6.13 + 2.95i)T + (25.5 - 32.0i)T^{2} \) |
| 43 | \( 1 + (-6.79 - 3.27i)T + (26.8 + 33.6i)T^{2} \) |
| 47 | \( 1 + (-2.17 + 5.53i)T + (-34.4 - 31.9i)T^{2} \) |
| 53 | \( 1 + (-4.53 + 4.88i)T + (-3.96 - 52.8i)T^{2} \) |
| 59 | \( 1 + (9.54 - 6.50i)T + (21.5 - 54.9i)T^{2} \) |
| 61 | \( 1 + (-2.17 - 2.34i)T + (-4.55 + 60.8i)T^{2} \) |
| 67 | \( 1 + (0.0184 - 0.0319i)T + (-33.5 - 58.0i)T^{2} \) |
| 71 | \( 1 + (1.83 + 0.419i)T + (63.9 + 30.8i)T^{2} \) |
| 73 | \( 1 + (0.0286 - 0.0112i)T + (53.5 - 49.6i)T^{2} \) |
| 79 | \( 1 + (2.78 + 4.83i)T + (-39.5 + 68.4i)T^{2} \) |
| 83 | \( 1 + (3.58 - 4.49i)T + (-18.4 - 80.9i)T^{2} \) |
| 89 | \( 1 + (-11.3 - 1.71i)T + (85.0 + 26.2i)T^{2} \) |
| 97 | \( 1 - 6.56iT - 97T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−13.43575665461268363728996405325, −12.23192653369166406784823585492, −11.35801247771508117766302011436, −10.34740631113832784029085865981, −9.126532581192878714100804938008, −7.19717907092941659797165787984, −6.32559455768236234904722474891, −5.51850928672678454253417426379, −4.59582613679478990405096326637, −2.35797645289887062936976819165,
2.00186897545887235060214785485, 4.34282485366941786310927935394, 5.03256459453736398753372425238, 5.78476349621207531353708080525, 7.45032170455981035791992601974, 9.152725470489791818496119993143, 10.21987863617256154377995703801, 11.10641418683533612044151730457, 12.40553020695001457597083778041, 12.69751194697449719796730496697
